Properties

Label 1024.2.a.h.1.4
Level $1024$
Weight $2$
Character 1024.1
Self dual yes
Analytic conductor $8.177$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.17668116698\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 512)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.84776\) of defining polynomial
Character \(\chi\) \(=\) 1024.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.61313 q^{3} -3.41421 q^{5} -1.53073 q^{7} +3.82843 q^{9} +O(q^{10})\) \(q+2.61313 q^{3} -3.41421 q^{5} -1.53073 q^{7} +3.82843 q^{9} -4.77791 q^{11} -0.585786 q^{13} -8.92177 q^{15} -2.82843 q^{17} -0.448342 q^{19} -4.00000 q^{21} -5.86030 q^{23} +6.65685 q^{25} +2.16478 q^{27} -4.58579 q^{29} +7.39104 q^{31} -12.4853 q^{33} +5.22625 q^{35} +5.07107 q^{37} -1.53073 q^{39} -4.00000 q^{41} -2.61313 q^{43} -13.0711 q^{45} +7.39104 q^{47} -4.65685 q^{49} -7.39104 q^{51} -7.41421 q^{53} +16.3128 q^{55} -1.17157 q^{57} +2.61313 q^{59} -13.0711 q^{61} -5.86030 q^{63} +2.00000 q^{65} -10.0042 q^{67} -15.3137 q^{69} +11.9832 q^{71} +10.4853 q^{73} +17.3952 q^{75} +7.31371 q^{77} +6.12293 q^{79} -5.82843 q^{81} +3.50981 q^{83} +9.65685 q^{85} -11.9832 q^{87} -0.828427 q^{89} +0.896683 q^{91} +19.3137 q^{93} +1.53073 q^{95} +10.8284 q^{97} -18.2919 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{5} + 4q^{9} + O(q^{10}) \) \( 4q - 8q^{5} + 4q^{9} - 8q^{13} - 16q^{21} + 4q^{25} - 24q^{29} - 16q^{33} - 8q^{37} - 16q^{41} - 24q^{45} + 4q^{49} - 24q^{53} - 16q^{57} - 24q^{61} + 8q^{65} - 16q^{69} + 8q^{73} - 16q^{77} - 12q^{81} + 16q^{85} + 8q^{89} + 32q^{93} + 32q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.61313 1.50869 0.754344 0.656479i \(-0.227955\pi\)
0.754344 + 0.656479i \(0.227955\pi\)
\(4\) 0 0
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0 0
\(7\) −1.53073 −0.578563 −0.289281 0.957244i \(-0.593416\pi\)
−0.289281 + 0.957244i \(0.593416\pi\)
\(8\) 0 0
\(9\) 3.82843 1.27614
\(10\) 0 0
\(11\) −4.77791 −1.44059 −0.720297 0.693666i \(-0.755995\pi\)
−0.720297 + 0.693666i \(0.755995\pi\)
\(12\) 0 0
\(13\) −0.585786 −0.162468 −0.0812340 0.996695i \(-0.525886\pi\)
−0.0812340 + 0.996695i \(0.525886\pi\)
\(14\) 0 0
\(15\) −8.92177 −2.30359
\(16\) 0 0
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) −0.448342 −0.102857 −0.0514283 0.998677i \(-0.516377\pi\)
−0.0514283 + 0.998677i \(0.516377\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) −5.86030 −1.22196 −0.610979 0.791647i \(-0.709224\pi\)
−0.610979 + 0.791647i \(0.709224\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 2.16478 0.416613
\(28\) 0 0
\(29\) −4.58579 −0.851559 −0.425780 0.904827i \(-0.640000\pi\)
−0.425780 + 0.904827i \(0.640000\pi\)
\(30\) 0 0
\(31\) 7.39104 1.32747 0.663735 0.747968i \(-0.268970\pi\)
0.663735 + 0.747968i \(0.268970\pi\)
\(32\) 0 0
\(33\) −12.4853 −2.17341
\(34\) 0 0
\(35\) 5.22625 0.883398
\(36\) 0 0
\(37\) 5.07107 0.833678 0.416839 0.908980i \(-0.363138\pi\)
0.416839 + 0.908980i \(0.363138\pi\)
\(38\) 0 0
\(39\) −1.53073 −0.245114
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −2.61313 −0.398498 −0.199249 0.979949i \(-0.563850\pi\)
−0.199249 + 0.979949i \(0.563850\pi\)
\(44\) 0 0
\(45\) −13.0711 −1.94852
\(46\) 0 0
\(47\) 7.39104 1.07809 0.539047 0.842276i \(-0.318785\pi\)
0.539047 + 0.842276i \(0.318785\pi\)
\(48\) 0 0
\(49\) −4.65685 −0.665265
\(50\) 0 0
\(51\) −7.39104 −1.03495
\(52\) 0 0
\(53\) −7.41421 −1.01842 −0.509210 0.860642i \(-0.670062\pi\)
−0.509210 + 0.860642i \(0.670062\pi\)
\(54\) 0 0
\(55\) 16.3128 2.19962
\(56\) 0 0
\(57\) −1.17157 −0.155179
\(58\) 0 0
\(59\) 2.61313 0.340200 0.170100 0.985427i \(-0.445591\pi\)
0.170100 + 0.985427i \(0.445591\pi\)
\(60\) 0 0
\(61\) −13.0711 −1.67358 −0.836789 0.547525i \(-0.815570\pi\)
−0.836789 + 0.547525i \(0.815570\pi\)
\(62\) 0 0
\(63\) −5.86030 −0.738329
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −10.0042 −1.22220 −0.611101 0.791552i \(-0.709273\pi\)
−0.611101 + 0.791552i \(0.709273\pi\)
\(68\) 0 0
\(69\) −15.3137 −1.84355
\(70\) 0 0
\(71\) 11.9832 1.42215 0.711074 0.703117i \(-0.248209\pi\)
0.711074 + 0.703117i \(0.248209\pi\)
\(72\) 0 0
\(73\) 10.4853 1.22721 0.613605 0.789613i \(-0.289719\pi\)
0.613605 + 0.789613i \(0.289719\pi\)
\(74\) 0 0
\(75\) 17.3952 2.00862
\(76\) 0 0
\(77\) 7.31371 0.833474
\(78\) 0 0
\(79\) 6.12293 0.688884 0.344442 0.938808i \(-0.388068\pi\)
0.344442 + 0.938808i \(0.388068\pi\)
\(80\) 0 0
\(81\) −5.82843 −0.647603
\(82\) 0 0
\(83\) 3.50981 0.385252 0.192626 0.981272i \(-0.438300\pi\)
0.192626 + 0.981272i \(0.438300\pi\)
\(84\) 0 0
\(85\) 9.65685 1.04743
\(86\) 0 0
\(87\) −11.9832 −1.28474
\(88\) 0 0
\(89\) −0.828427 −0.0878131 −0.0439065 0.999036i \(-0.513980\pi\)
−0.0439065 + 0.999036i \(0.513980\pi\)
\(90\) 0 0
\(91\) 0.896683 0.0939979
\(92\) 0 0
\(93\) 19.3137 2.00274
\(94\) 0 0
\(95\) 1.53073 0.157050
\(96\) 0 0
\(97\) 10.8284 1.09946 0.549730 0.835342i \(-0.314731\pi\)
0.549730 + 0.835342i \(0.314731\pi\)
\(98\) 0 0
\(99\) −18.2919 −1.83840
\(100\) 0 0
\(101\) −2.92893 −0.291440 −0.145720 0.989326i \(-0.546550\pi\)
−0.145720 + 0.989326i \(0.546550\pi\)
\(102\) 0 0
\(103\) −2.79884 −0.275777 −0.137889 0.990448i \(-0.544032\pi\)
−0.137889 + 0.990448i \(0.544032\pi\)
\(104\) 0 0
\(105\) 13.6569 1.33277
\(106\) 0 0
\(107\) 10.0042 0.967139 0.483569 0.875306i \(-0.339340\pi\)
0.483569 + 0.875306i \(0.339340\pi\)
\(108\) 0 0
\(109\) −1.07107 −0.102590 −0.0512948 0.998684i \(-0.516335\pi\)
−0.0512948 + 0.998684i \(0.516335\pi\)
\(110\) 0 0
\(111\) 13.2513 1.25776
\(112\) 0 0
\(113\) 3.65685 0.344008 0.172004 0.985096i \(-0.444976\pi\)
0.172004 + 0.985096i \(0.444976\pi\)
\(114\) 0 0
\(115\) 20.0083 1.86579
\(116\) 0 0
\(117\) −2.24264 −0.207332
\(118\) 0 0
\(119\) 4.32957 0.396891
\(120\) 0 0
\(121\) 11.8284 1.07531
\(122\) 0 0
\(123\) −10.4525 −0.942471
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −13.5140 −1.19917 −0.599586 0.800311i \(-0.704668\pi\)
−0.599586 + 0.800311i \(0.704668\pi\)
\(128\) 0 0
\(129\) −6.82843 −0.601209
\(130\) 0 0
\(131\) 9.10748 0.795724 0.397862 0.917445i \(-0.369752\pi\)
0.397862 + 0.917445i \(0.369752\pi\)
\(132\) 0 0
\(133\) 0.686292 0.0595090
\(134\) 0 0
\(135\) −7.39104 −0.636119
\(136\) 0 0
\(137\) −20.9706 −1.79164 −0.895818 0.444421i \(-0.853409\pi\)
−0.895818 + 0.444421i \(0.853409\pi\)
\(138\) 0 0
\(139\) −19.1886 −1.62755 −0.813776 0.581178i \(-0.802592\pi\)
−0.813776 + 0.581178i \(0.802592\pi\)
\(140\) 0 0
\(141\) 19.3137 1.62651
\(142\) 0 0
\(143\) 2.79884 0.234050
\(144\) 0 0
\(145\) 15.6569 1.30023
\(146\) 0 0
\(147\) −12.1689 −1.00368
\(148\) 0 0
\(149\) −10.2426 −0.839110 −0.419555 0.907730i \(-0.637814\pi\)
−0.419555 + 0.907730i \(0.637814\pi\)
\(150\) 0 0
\(151\) 8.92177 0.726043 0.363022 0.931781i \(-0.381745\pi\)
0.363022 + 0.931781i \(0.381745\pi\)
\(152\) 0 0
\(153\) −10.8284 −0.875426
\(154\) 0 0
\(155\) −25.2346 −2.02689
\(156\) 0 0
\(157\) 6.24264 0.498217 0.249108 0.968476i \(-0.419862\pi\)
0.249108 + 0.968476i \(0.419862\pi\)
\(158\) 0 0
\(159\) −19.3743 −1.53648
\(160\) 0 0
\(161\) 8.97056 0.706979
\(162\) 0 0
\(163\) 13.0656 1.02338 0.511690 0.859170i \(-0.329020\pi\)
0.511690 + 0.859170i \(0.329020\pi\)
\(164\) 0 0
\(165\) 42.6274 3.31854
\(166\) 0 0
\(167\) −1.53073 −0.118452 −0.0592259 0.998245i \(-0.518863\pi\)
−0.0592259 + 0.998245i \(0.518863\pi\)
\(168\) 0 0
\(169\) −12.6569 −0.973604
\(170\) 0 0
\(171\) −1.71644 −0.131260
\(172\) 0 0
\(173\) −23.2132 −1.76487 −0.882434 0.470437i \(-0.844096\pi\)
−0.882434 + 0.470437i \(0.844096\pi\)
\(174\) 0 0
\(175\) −10.1899 −0.770282
\(176\) 0 0
\(177\) 6.82843 0.513256
\(178\) 0 0
\(179\) 1.71644 0.128293 0.0641465 0.997940i \(-0.479568\pi\)
0.0641465 + 0.997940i \(0.479568\pi\)
\(180\) 0 0
\(181\) 5.75736 0.427941 0.213971 0.976840i \(-0.431360\pi\)
0.213971 + 0.976840i \(0.431360\pi\)
\(182\) 0 0
\(183\) −34.1563 −2.52491
\(184\) 0 0
\(185\) −17.3137 −1.27293
\(186\) 0 0
\(187\) 13.5140 0.988239
\(188\) 0 0
\(189\) −3.31371 −0.241037
\(190\) 0 0
\(191\) 13.5140 0.977837 0.488918 0.872330i \(-0.337392\pi\)
0.488918 + 0.872330i \(0.337392\pi\)
\(192\) 0 0
\(193\) −16.4853 −1.18664 −0.593318 0.804968i \(-0.702182\pi\)
−0.593318 + 0.804968i \(0.702182\pi\)
\(194\) 0 0
\(195\) 5.22625 0.374260
\(196\) 0 0
\(197\) −14.7279 −1.04932 −0.524660 0.851312i \(-0.675808\pi\)
−0.524660 + 0.851312i \(0.675808\pi\)
\(198\) 0 0
\(199\) −22.4357 −1.59043 −0.795214 0.606329i \(-0.792641\pi\)
−0.795214 + 0.606329i \(0.792641\pi\)
\(200\) 0 0
\(201\) −26.1421 −1.84392
\(202\) 0 0
\(203\) 7.01962 0.492681
\(204\) 0 0
\(205\) 13.6569 0.953836
\(206\) 0 0
\(207\) −22.4357 −1.55939
\(208\) 0 0
\(209\) 2.14214 0.148175
\(210\) 0 0
\(211\) 26.9510 1.85538 0.927692 0.373346i \(-0.121789\pi\)
0.927692 + 0.373346i \(0.121789\pi\)
\(212\) 0 0
\(213\) 31.3137 2.14558
\(214\) 0 0
\(215\) 8.92177 0.608460
\(216\) 0 0
\(217\) −11.3137 −0.768025
\(218\) 0 0
\(219\) 27.3994 1.85148
\(220\) 0 0
\(221\) 1.65685 0.111452
\(222\) 0 0
\(223\) 7.39104 0.494940 0.247470 0.968896i \(-0.420401\pi\)
0.247470 + 0.968896i \(0.420401\pi\)
\(224\) 0 0
\(225\) 25.4853 1.69902
\(226\) 0 0
\(227\) 2.61313 0.173439 0.0867196 0.996233i \(-0.472362\pi\)
0.0867196 + 0.996233i \(0.472362\pi\)
\(228\) 0 0
\(229\) 15.8995 1.05067 0.525334 0.850896i \(-0.323940\pi\)
0.525334 + 0.850896i \(0.323940\pi\)
\(230\) 0 0
\(231\) 19.1116 1.25745
\(232\) 0 0
\(233\) 15.1716 0.993923 0.496961 0.867773i \(-0.334449\pi\)
0.496961 + 0.867773i \(0.334449\pi\)
\(234\) 0 0
\(235\) −25.2346 −1.64612
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) −29.5641 −1.91235 −0.956173 0.292803i \(-0.905412\pi\)
−0.956173 + 0.292803i \(0.905412\pi\)
\(240\) 0 0
\(241\) −18.8284 −1.21285 −0.606423 0.795142i \(-0.707396\pi\)
−0.606423 + 0.795142i \(0.707396\pi\)
\(242\) 0 0
\(243\) −21.7248 −1.39364
\(244\) 0 0
\(245\) 15.8995 1.01578
\(246\) 0 0
\(247\) 0.262632 0.0167109
\(248\) 0 0
\(249\) 9.17157 0.581225
\(250\) 0 0
\(251\) −13.9623 −0.881293 −0.440647 0.897681i \(-0.645251\pi\)
−0.440647 + 0.897681i \(0.645251\pi\)
\(252\) 0 0
\(253\) 28.0000 1.76034
\(254\) 0 0
\(255\) 25.2346 1.58025
\(256\) 0 0
\(257\) 18.9706 1.18335 0.591676 0.806176i \(-0.298467\pi\)
0.591676 + 0.806176i \(0.298467\pi\)
\(258\) 0 0
\(259\) −7.76245 −0.482335
\(260\) 0 0
\(261\) −17.5563 −1.08671
\(262\) 0 0
\(263\) 20.6424 1.27286 0.636432 0.771333i \(-0.280410\pi\)
0.636432 + 0.771333i \(0.280410\pi\)
\(264\) 0 0
\(265\) 25.3137 1.55501
\(266\) 0 0
\(267\) −2.16478 −0.132483
\(268\) 0 0
\(269\) 5.55635 0.338777 0.169388 0.985549i \(-0.445821\pi\)
0.169388 + 0.985549i \(0.445821\pi\)
\(270\) 0 0
\(271\) 28.2960 1.71886 0.859431 0.511252i \(-0.170818\pi\)
0.859431 + 0.511252i \(0.170818\pi\)
\(272\) 0 0
\(273\) 2.34315 0.141814
\(274\) 0 0
\(275\) −31.8059 −1.91797
\(276\) 0 0
\(277\) 0.585786 0.0351965 0.0175982 0.999845i \(-0.494398\pi\)
0.0175982 + 0.999845i \(0.494398\pi\)
\(278\) 0 0
\(279\) 28.2960 1.69404
\(280\) 0 0
\(281\) −21.7990 −1.30042 −0.650209 0.759755i \(-0.725319\pi\)
−0.650209 + 0.759755i \(0.725319\pi\)
\(282\) 0 0
\(283\) 9.10748 0.541383 0.270692 0.962666i \(-0.412748\pi\)
0.270692 + 0.962666i \(0.412748\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 6.12293 0.361425
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 28.2960 1.65874
\(292\) 0 0
\(293\) −3.41421 −0.199460 −0.0997302 0.995015i \(-0.531798\pi\)
−0.0997302 + 0.995015i \(0.531798\pi\)
\(294\) 0 0
\(295\) −8.92177 −0.519446
\(296\) 0 0
\(297\) −10.3431 −0.600170
\(298\) 0 0
\(299\) 3.43289 0.198529
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) −7.65367 −0.439692
\(304\) 0 0
\(305\) 44.6274 2.55536
\(306\) 0 0
\(307\) −25.6829 −1.46580 −0.732901 0.680336i \(-0.761834\pi\)
−0.732901 + 0.680336i \(0.761834\pi\)
\(308\) 0 0
\(309\) −7.31371 −0.416062
\(310\) 0 0
\(311\) −7.12840 −0.404215 −0.202107 0.979363i \(-0.564779\pi\)
−0.202107 + 0.979363i \(0.564779\pi\)
\(312\) 0 0
\(313\) −8.68629 −0.490978 −0.245489 0.969399i \(-0.578949\pi\)
−0.245489 + 0.969399i \(0.578949\pi\)
\(314\) 0 0
\(315\) 20.0083 1.12734
\(316\) 0 0
\(317\) −12.5858 −0.706888 −0.353444 0.935456i \(-0.614990\pi\)
−0.353444 + 0.935456i \(0.614990\pi\)
\(318\) 0 0
\(319\) 21.9105 1.22675
\(320\) 0 0
\(321\) 26.1421 1.45911
\(322\) 0 0
\(323\) 1.26810 0.0705590
\(324\) 0 0
\(325\) −3.89949 −0.216305
\(326\) 0 0
\(327\) −2.79884 −0.154776
\(328\) 0 0
\(329\) −11.3137 −0.623745
\(330\) 0 0
\(331\) 5.67459 0.311904 0.155952 0.987765i \(-0.450156\pi\)
0.155952 + 0.987765i \(0.450156\pi\)
\(332\) 0 0
\(333\) 19.4142 1.06389
\(334\) 0 0
\(335\) 34.1563 1.86616
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) 9.55582 0.519001
\(340\) 0 0
\(341\) −35.3137 −1.91234
\(342\) 0 0
\(343\) 17.8435 0.963461
\(344\) 0 0
\(345\) 52.2843 2.81489
\(346\) 0 0
\(347\) 21.7248 1.16625 0.583123 0.812384i \(-0.301830\pi\)
0.583123 + 0.812384i \(0.301830\pi\)
\(348\) 0 0
\(349\) 22.7279 1.21660 0.608299 0.793708i \(-0.291852\pi\)
0.608299 + 0.793708i \(0.291852\pi\)
\(350\) 0 0
\(351\) −1.26810 −0.0676862
\(352\) 0 0
\(353\) −10.9706 −0.583904 −0.291952 0.956433i \(-0.594305\pi\)
−0.291952 + 0.956433i \(0.594305\pi\)
\(354\) 0 0
\(355\) −40.9133 −2.17145
\(356\) 0 0
\(357\) 11.3137 0.598785
\(358\) 0 0
\(359\) −8.92177 −0.470873 −0.235437 0.971890i \(-0.575652\pi\)
−0.235437 + 0.971890i \(0.575652\pi\)
\(360\) 0 0
\(361\) −18.7990 −0.989421
\(362\) 0 0
\(363\) 30.9092 1.62231
\(364\) 0 0
\(365\) −35.7990 −1.87380
\(366\) 0 0
\(367\) −22.1731 −1.15743 −0.578713 0.815531i \(-0.696445\pi\)
−0.578713 + 0.815531i \(0.696445\pi\)
\(368\) 0 0
\(369\) −15.3137 −0.797200
\(370\) 0 0
\(371\) 11.3492 0.589220
\(372\) 0 0
\(373\) −18.7279 −0.969695 −0.484848 0.874599i \(-0.661125\pi\)
−0.484848 + 0.874599i \(0.661125\pi\)
\(374\) 0 0
\(375\) −14.7821 −0.763343
\(376\) 0 0
\(377\) 2.68629 0.138351
\(378\) 0 0
\(379\) 4.40649 0.226346 0.113173 0.993575i \(-0.463899\pi\)
0.113173 + 0.993575i \(0.463899\pi\)
\(380\) 0 0
\(381\) −35.3137 −1.80918
\(382\) 0 0
\(383\) −8.65914 −0.442461 −0.221231 0.975222i \(-0.571007\pi\)
−0.221231 + 0.975222i \(0.571007\pi\)
\(384\) 0 0
\(385\) −24.9706 −1.27262
\(386\) 0 0
\(387\) −10.0042 −0.508540
\(388\) 0 0
\(389\) 16.3848 0.830741 0.415371 0.909652i \(-0.363652\pi\)
0.415371 + 0.909652i \(0.363652\pi\)
\(390\) 0 0
\(391\) 16.5754 0.838256
\(392\) 0 0
\(393\) 23.7990 1.20050
\(394\) 0 0
\(395\) −20.9050 −1.05185
\(396\) 0 0
\(397\) −9.07107 −0.455264 −0.227632 0.973747i \(-0.573098\pi\)
−0.227632 + 0.973747i \(0.573098\pi\)
\(398\) 0 0
\(399\) 1.79337 0.0897806
\(400\) 0 0
\(401\) 24.4853 1.22274 0.611368 0.791346i \(-0.290619\pi\)
0.611368 + 0.791346i \(0.290619\pi\)
\(402\) 0 0
\(403\) −4.32957 −0.215671
\(404\) 0 0
\(405\) 19.8995 0.988814
\(406\) 0 0
\(407\) −24.2291 −1.20099
\(408\) 0 0
\(409\) −8.68629 −0.429509 −0.214755 0.976668i \(-0.568895\pi\)
−0.214755 + 0.976668i \(0.568895\pi\)
\(410\) 0 0
\(411\) −54.7987 −2.70302
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) −11.9832 −0.588234
\(416\) 0 0
\(417\) −50.1421 −2.45547
\(418\) 0 0
\(419\) −14.3337 −0.700249 −0.350124 0.936703i \(-0.613861\pi\)
−0.350124 + 0.936703i \(0.613861\pi\)
\(420\) 0 0
\(421\) −11.4142 −0.556295 −0.278147 0.960538i \(-0.589720\pi\)
−0.278147 + 0.960538i \(0.589720\pi\)
\(422\) 0 0
\(423\) 28.2960 1.37580
\(424\) 0 0
\(425\) −18.8284 −0.913313
\(426\) 0 0
\(427\) 20.0083 0.968271
\(428\) 0 0
\(429\) 7.31371 0.353109
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −25.4558 −1.22333 −0.611665 0.791117i \(-0.709500\pi\)
−0.611665 + 0.791117i \(0.709500\pi\)
\(434\) 0 0
\(435\) 40.9133 1.96164
\(436\) 0 0
\(437\) 2.62742 0.125686
\(438\) 0 0
\(439\) 18.8490 0.899614 0.449807 0.893126i \(-0.351493\pi\)
0.449807 + 0.893126i \(0.351493\pi\)
\(440\) 0 0
\(441\) −17.8284 −0.848973
\(442\) 0 0
\(443\) −10.0042 −0.475312 −0.237656 0.971349i \(-0.576379\pi\)
−0.237656 + 0.971349i \(0.576379\pi\)
\(444\) 0 0
\(445\) 2.82843 0.134080
\(446\) 0 0
\(447\) −26.7653 −1.26596
\(448\) 0 0
\(449\) 22.1421 1.04495 0.522476 0.852654i \(-0.325008\pi\)
0.522476 + 0.852654i \(0.325008\pi\)
\(450\) 0 0
\(451\) 19.1116 0.899932
\(452\) 0 0
\(453\) 23.3137 1.09537
\(454\) 0 0
\(455\) −3.06147 −0.143524
\(456\) 0 0
\(457\) 18.6274 0.871354 0.435677 0.900103i \(-0.356509\pi\)
0.435677 + 0.900103i \(0.356509\pi\)
\(458\) 0 0
\(459\) −6.12293 −0.285794
\(460\) 0 0
\(461\) −29.7574 −1.38594 −0.692969 0.720967i \(-0.743698\pi\)
−0.692969 + 0.720967i \(0.743698\pi\)
\(462\) 0 0
\(463\) 2.53620 0.117867 0.0589337 0.998262i \(-0.481230\pi\)
0.0589337 + 0.998262i \(0.481230\pi\)
\(464\) 0 0
\(465\) −65.9411 −3.05795
\(466\) 0 0
\(467\) 17.9205 0.829260 0.414630 0.909990i \(-0.363911\pi\)
0.414630 + 0.909990i \(0.363911\pi\)
\(468\) 0 0
\(469\) 15.3137 0.707121
\(470\) 0 0
\(471\) 16.3128 0.751654
\(472\) 0 0
\(473\) 12.4853 0.574074
\(474\) 0 0
\(475\) −2.98454 −0.136940
\(476\) 0 0
\(477\) −28.3848 −1.29965
\(478\) 0 0
\(479\) −40.5419 −1.85241 −0.926204 0.377024i \(-0.876948\pi\)
−0.926204 + 0.377024i \(0.876948\pi\)
\(480\) 0 0
\(481\) −2.97056 −0.135446
\(482\) 0 0
\(483\) 23.4412 1.06661
\(484\) 0 0
\(485\) −36.9706 −1.67875
\(486\) 0 0
\(487\) −37.2178 −1.68650 −0.843250 0.537521i \(-0.819361\pi\)
−0.843250 + 0.537521i \(0.819361\pi\)
\(488\) 0 0
\(489\) 34.1421 1.54396
\(490\) 0 0
\(491\) −17.3952 −0.785034 −0.392517 0.919745i \(-0.628396\pi\)
−0.392517 + 0.919745i \(0.628396\pi\)
\(492\) 0 0
\(493\) 12.9706 0.584165
\(494\) 0 0
\(495\) 62.4524 2.80703
\(496\) 0 0
\(497\) −18.3431 −0.822803
\(498\) 0 0
\(499\) 16.4985 0.738575 0.369287 0.929315i \(-0.379602\pi\)
0.369287 + 0.929315i \(0.379602\pi\)
\(500\) 0 0
\(501\) −4.00000 −0.178707
\(502\) 0 0
\(503\) −18.1062 −0.807314 −0.403657 0.914910i \(-0.632261\pi\)
−0.403657 + 0.914910i \(0.632261\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) −33.0740 −1.46887
\(508\) 0 0
\(509\) −17.7574 −0.787081 −0.393541 0.919307i \(-0.628750\pi\)
−0.393541 + 0.919307i \(0.628750\pi\)
\(510\) 0 0
\(511\) −16.0502 −0.710018
\(512\) 0 0
\(513\) −0.970563 −0.0428514
\(514\) 0 0
\(515\) 9.55582 0.421080
\(516\) 0 0
\(517\) −35.3137 −1.55310
\(518\) 0 0
\(519\) −60.6590 −2.66264
\(520\) 0 0
\(521\) 17.6569 0.773561 0.386780 0.922172i \(-0.373587\pi\)
0.386780 + 0.922172i \(0.373587\pi\)
\(522\) 0 0
\(523\) 39.5683 1.73020 0.865101 0.501598i \(-0.167254\pi\)
0.865101 + 0.501598i \(0.167254\pi\)
\(524\) 0 0
\(525\) −26.6274 −1.16212
\(526\) 0 0
\(527\) −20.9050 −0.910636
\(528\) 0 0
\(529\) 11.3431 0.493180
\(530\) 0 0
\(531\) 10.0042 0.434144
\(532\) 0 0
\(533\) 2.34315 0.101493
\(534\) 0 0
\(535\) −34.1563 −1.47671
\(536\) 0 0
\(537\) 4.48528 0.193554
\(538\) 0 0
\(539\) 22.2500 0.958377
\(540\) 0 0
\(541\) −1.75736 −0.0755548 −0.0377774 0.999286i \(-0.512028\pi\)
−0.0377774 + 0.999286i \(0.512028\pi\)
\(542\) 0 0
\(543\) 15.0447 0.645630
\(544\) 0 0
\(545\) 3.65685 0.156642
\(546\) 0 0
\(547\) −1.34502 −0.0575091 −0.0287545 0.999587i \(-0.509154\pi\)
−0.0287545 + 0.999587i \(0.509154\pi\)
\(548\) 0 0
\(549\) −50.0416 −2.13572
\(550\) 0 0
\(551\) 2.05600 0.0875885
\(552\) 0 0
\(553\) −9.37258 −0.398563
\(554\) 0 0
\(555\) −45.2429 −1.92045
\(556\) 0 0
\(557\) −8.58579 −0.363791 −0.181896 0.983318i \(-0.558223\pi\)
−0.181896 + 0.983318i \(0.558223\pi\)
\(558\) 0 0
\(559\) 1.53073 0.0647431
\(560\) 0 0
\(561\) 35.3137 1.49095
\(562\) 0 0
\(563\) −21.3533 −0.899936 −0.449968 0.893045i \(-0.648565\pi\)
−0.449968 + 0.893045i \(0.648565\pi\)
\(564\) 0 0
\(565\) −12.4853 −0.525260
\(566\) 0 0
\(567\) 8.92177 0.374679
\(568\) 0 0
\(569\) −14.3431 −0.601296 −0.300648 0.953735i \(-0.597203\pi\)
−0.300648 + 0.953735i \(0.597203\pi\)
\(570\) 0 0
\(571\) −22.2500 −0.931135 −0.465567 0.885012i \(-0.654150\pi\)
−0.465567 + 0.885012i \(0.654150\pi\)
\(572\) 0 0
\(573\) 35.3137 1.47525
\(574\) 0 0
\(575\) −39.0112 −1.62688
\(576\) 0 0
\(577\) −5.31371 −0.221213 −0.110606 0.993864i \(-0.535279\pi\)
−0.110606 + 0.993864i \(0.535279\pi\)
\(578\) 0 0
\(579\) −43.0781 −1.79027
\(580\) 0 0
\(581\) −5.37258 −0.222892
\(582\) 0 0
\(583\) 35.4244 1.46713
\(584\) 0 0
\(585\) 7.65685 0.316572
\(586\) 0 0
\(587\) 29.1158 1.20174 0.600869 0.799348i \(-0.294821\pi\)
0.600869 + 0.799348i \(0.294821\pi\)
\(588\) 0 0
\(589\) −3.31371 −0.136539
\(590\) 0 0
\(591\) −38.4859 −1.58310
\(592\) 0 0
\(593\) −13.3137 −0.546728 −0.273364 0.961911i \(-0.588136\pi\)
−0.273364 + 0.961911i \(0.588136\pi\)
\(594\) 0 0
\(595\) −14.7821 −0.606006
\(596\) 0 0
\(597\) −58.6274 −2.39946
\(598\) 0 0
\(599\) −15.7875 −0.645061 −0.322531 0.946559i \(-0.604534\pi\)
−0.322531 + 0.946559i \(0.604534\pi\)
\(600\) 0 0
\(601\) 15.1716 0.618861 0.309431 0.950922i \(-0.399862\pi\)
0.309431 + 0.950922i \(0.399862\pi\)
\(602\) 0 0
\(603\) −38.3002 −1.55970
\(604\) 0 0
\(605\) −40.3848 −1.64187
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 18.3431 0.743302
\(610\) 0 0
\(611\) −4.32957 −0.175156
\(612\) 0 0
\(613\) 36.5858 1.47769 0.738843 0.673878i \(-0.235373\pi\)
0.738843 + 0.673878i \(0.235373\pi\)
\(614\) 0 0
\(615\) 35.6871 1.43904
\(616\) 0 0
\(617\) 21.5147 0.866150 0.433075 0.901358i \(-0.357429\pi\)
0.433075 + 0.901358i \(0.357429\pi\)
\(618\) 0 0
\(619\) −4.77791 −0.192040 −0.0960202 0.995379i \(-0.530611\pi\)
−0.0960202 + 0.995379i \(0.530611\pi\)
\(620\) 0 0
\(621\) −12.6863 −0.509083
\(622\) 0 0
\(623\) 1.26810 0.0508054
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) 5.59767 0.223549
\(628\) 0 0
\(629\) −14.3431 −0.571899
\(630\) 0 0
\(631\) 2.79884 0.111420 0.0557099 0.998447i \(-0.482258\pi\)
0.0557099 + 0.998447i \(0.482258\pi\)
\(632\) 0 0
\(633\) 70.4264 2.79920
\(634\) 0 0
\(635\) 46.1396 1.83099
\(636\) 0 0
\(637\) 2.72792 0.108084
\(638\) 0 0
\(639\) 45.8770 1.81486
\(640\) 0 0
\(641\) −9.85786 −0.389362 −0.194681 0.980867i \(-0.562367\pi\)
−0.194681 + 0.980867i \(0.562367\pi\)
\(642\) 0 0
\(643\) −1.34502 −0.0530426 −0.0265213 0.999648i \(-0.508443\pi\)
−0.0265213 + 0.999648i \(0.508443\pi\)
\(644\) 0 0
\(645\) 23.3137 0.917976
\(646\) 0 0
\(647\) −0.262632 −0.0103251 −0.00516257 0.999987i \(-0.501643\pi\)
−0.00516257 + 0.999987i \(0.501643\pi\)
\(648\) 0 0
\(649\) −12.4853 −0.490090
\(650\) 0 0
\(651\) −29.5641 −1.15871
\(652\) 0 0
\(653\) 37.5563 1.46969 0.734847 0.678233i \(-0.237254\pi\)
0.734847 + 0.678233i \(0.237254\pi\)
\(654\) 0 0
\(655\) −31.0949 −1.21498
\(656\) 0 0
\(657\) 40.1421 1.56609
\(658\) 0 0
\(659\) 33.0740 1.28838 0.644189 0.764866i \(-0.277195\pi\)
0.644189 + 0.764866i \(0.277195\pi\)
\(660\) 0 0
\(661\) −14.9289 −0.580668 −0.290334 0.956925i \(-0.593766\pi\)
−0.290334 + 0.956925i \(0.593766\pi\)
\(662\) 0 0
\(663\) 4.32957 0.168147
\(664\) 0 0
\(665\) −2.34315 −0.0908633
\(666\) 0 0
\(667\) 26.8741 1.04057
\(668\) 0 0
\(669\) 19.3137 0.746711
\(670\) 0 0
\(671\) 62.4524 2.41095
\(672\) 0 0
\(673\) −21.1716 −0.816104 −0.408052 0.912959i \(-0.633792\pi\)
−0.408052 + 0.912959i \(0.633792\pi\)
\(674\) 0 0
\(675\) 14.4107 0.554666
\(676\) 0 0
\(677\) −17.5563 −0.674745 −0.337373 0.941371i \(-0.609538\pi\)
−0.337373 + 0.941371i \(0.609538\pi\)
\(678\) 0 0
\(679\) −16.5754 −0.636107
\(680\) 0 0
\(681\) 6.82843 0.261666
\(682\) 0 0
\(683\) 20.0852 0.768541 0.384270 0.923221i \(-0.374453\pi\)
0.384270 + 0.923221i \(0.374453\pi\)
\(684\) 0 0
\(685\) 71.5980 2.73562
\(686\) 0 0
\(687\) 41.5474 1.58513
\(688\) 0 0
\(689\) 4.34315 0.165461
\(690\) 0 0
\(691\) −19.9314 −0.758226 −0.379113 0.925350i \(-0.623771\pi\)
−0.379113 + 0.925350i \(0.623771\pi\)
\(692\) 0 0
\(693\) 28.0000 1.06363
\(694\) 0 0
\(695\) 65.5139 2.48508
\(696\) 0 0
\(697\) 11.3137 0.428537
\(698\) 0 0
\(699\) 39.6452 1.49952
\(700\) 0 0
\(701\) 28.8701 1.09041 0.545204 0.838304i \(-0.316452\pi\)
0.545204 + 0.838304i \(0.316452\pi\)
\(702\) 0 0
\(703\) −2.27357 −0.0857493
\(704\) 0 0
\(705\) −65.9411 −2.48349
\(706\) 0 0
\(707\) 4.48342 0.168616
\(708\) 0 0
\(709\) 35.6985 1.34068 0.670342 0.742052i \(-0.266147\pi\)
0.670342 + 0.742052i \(0.266147\pi\)
\(710\) 0 0
\(711\) 23.4412 0.879114
\(712\) 0 0
\(713\) −43.3137 −1.62211
\(714\) 0 0
\(715\) −9.55582 −0.357367
\(716\) 0 0
\(717\) −77.2548 −2.88513
\(718\) 0 0
\(719\) 12.2459 0.456694 0.228347 0.973580i \(-0.426668\pi\)
0.228347 + 0.973580i \(0.426668\pi\)
\(720\) 0 0
\(721\) 4.28427 0.159555
\(722\) 0 0
\(723\) −49.2011 −1.82981
\(724\) 0 0
\(725\) −30.5269 −1.13374
\(726\) 0 0
\(727\) −42.8155 −1.58794 −0.793969 0.607958i \(-0.791989\pi\)
−0.793969 + 0.607958i \(0.791989\pi\)
\(728\) 0 0
\(729\) −39.2843 −1.45497
\(730\) 0 0
\(731\) 7.39104 0.273367
\(732\) 0 0
\(733\) −46.5269 −1.71851 −0.859255 0.511547i \(-0.829073\pi\)
−0.859255 + 0.511547i \(0.829073\pi\)
\(734\) 0 0
\(735\) 41.5474 1.53250
\(736\) 0 0
\(737\) 47.7990 1.76070
\(738\) 0 0
\(739\) 31.4344 1.15633 0.578167 0.815918i \(-0.303768\pi\)
0.578167 + 0.815918i \(0.303768\pi\)
\(740\) 0 0
\(741\) 0.686292 0.0252115
\(742\) 0 0
\(743\) −28.5587 −1.04772 −0.523858 0.851806i \(-0.675508\pi\)
−0.523858 + 0.851806i \(0.675508\pi\)
\(744\) 0 0
\(745\) 34.9706 1.28122
\(746\) 0 0
\(747\) 13.4370 0.491636
\(748\) 0 0
\(749\) −15.3137 −0.559551
\(750\) 0 0
\(751\) −6.12293 −0.223429 −0.111715 0.993740i \(-0.535634\pi\)
−0.111715 + 0.993740i \(0.535634\pi\)
\(752\) 0 0
\(753\) −36.4853 −1.32960
\(754\) 0 0
\(755\) −30.4608 −1.10858
\(756\) 0 0
\(757\) 33.0711 1.20199 0.600994 0.799253i \(-0.294771\pi\)
0.600994 + 0.799253i \(0.294771\pi\)
\(758\) 0 0
\(759\) 73.1675 2.65581
\(760\) 0 0
\(761\) −10.6274 −0.385244 −0.192622 0.981273i \(-0.561699\pi\)
−0.192622 + 0.981273i \(0.561699\pi\)
\(762\) 0 0
\(763\) 1.63952 0.0593546
\(764\) 0 0
\(765\) 36.9706 1.33667
\(766\) 0 0
\(767\) −1.53073 −0.0552716
\(768\) 0 0
\(769\) 10.8284 0.390483 0.195242 0.980755i \(-0.437451\pi\)
0.195242 + 0.980755i \(0.437451\pi\)
\(770\) 0 0
\(771\) 49.5725 1.78531
\(772\) 0 0
\(773\) 19.6985 0.708505 0.354253 0.935150i \(-0.384735\pi\)
0.354253 + 0.935150i \(0.384735\pi\)
\(774\) 0 0
\(775\) 49.2011 1.76735
\(776\) 0 0
\(777\) −20.2843 −0.727694
\(778\) 0 0
\(779\) 1.79337 0.0642540
\(780\) 0 0
\(781\) −57.2548 −2.04874
\(782\) 0 0
\(783\) −9.92724 −0.354771
\(784\) 0 0
\(785\) −21.3137 −0.760719
\(786\) 0 0
\(787\) −48.7527 −1.73785 −0.868923 0.494947i \(-0.835187\pi\)
−0.868923 + 0.494947i \(0.835187\pi\)
\(788\) 0 0
\(789\) 53.9411 1.92035
\(790\) 0 0
\(791\) −5.59767 −0.199030
\(792\) 0 0
\(793\) 7.65685 0.271903
\(794\) 0 0
\(795\) 66.1479 2.34602
\(796\) 0 0
\(797\) −19.6985 −0.697756 −0.348878 0.937168i \(-0.613437\pi\)
−0.348878 + 0.937168i \(0.613437\pi\)
\(798\) 0 0
\(799\) −20.9050 −0.739566
\(800\) 0 0
\(801\) −3.17157 −0.112062
\(802\) 0 0
\(803\) −50.0977 −1.76791
\(804\) 0 0
\(805\) −30.6274 −1.07947
\(806\) 0 0
\(807\) 14.5194 0.511108
\(808\) 0 0
\(809\) −47.3137 −1.66346 −0.831731 0.555179i \(-0.812650\pi\)
−0.831731 + 0.555179i \(0.812650\pi\)
\(810\) 0 0
\(811\) 20.4567 0.718331 0.359165 0.933274i \(-0.383061\pi\)
0.359165 + 0.933274i \(0.383061\pi\)
\(812\) 0 0
\(813\) 73.9411 2.59323
\(814\) 0 0
\(815\) −44.6088 −1.56258
\(816\) 0 0
\(817\) 1.17157 0.0409881
\(818\) 0 0
\(819\) 3.43289 0.119955
\(820\) 0 0
\(821\) 26.5269 0.925796 0.462898 0.886412i \(-0.346810\pi\)
0.462898 + 0.886412i \(0.346810\pi\)
\(822\) 0 0
\(823\) −42.8155 −1.49245 −0.746227 0.665692i \(-0.768137\pi\)
−0.746227 + 0.665692i \(0.768137\pi\)
\(824\) 0 0
\(825\) −83.1127 −2.89361
\(826\) 0 0
\(827\) −29.1158 −1.01246 −0.506228 0.862400i \(-0.668961\pi\)
−0.506228 + 0.862400i \(0.668961\pi\)
\(828\) 0 0
\(829\) −24.3848 −0.846918 −0.423459 0.905915i \(-0.639184\pi\)
−0.423459 + 0.905915i \(0.639184\pi\)
\(830\) 0 0
\(831\) 1.53073 0.0531006
\(832\) 0 0
\(833\) 13.1716 0.456368
\(834\) 0 0
\(835\) 5.22625 0.180862
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) 0 0
\(839\) −39.7540 −1.37246 −0.686231 0.727384i \(-0.740736\pi\)
−0.686231 + 0.727384i \(0.740736\pi\)
\(840\) 0 0
\(841\) −7.97056 −0.274847
\(842\) 0 0
\(843\) −56.9635 −1.96193
\(844\) 0 0
\(845\) 43.2132 1.48658
\(846\) 0 0
\(847\) −18.1062 −0.622135
\(848\) 0 0
\(849\) 23.7990 0.816779
\(850\) 0 0
\(851\) −29.7180 −1.01872
\(852\) 0 0
\(853\) 51.8995 1.77700 0.888502 0.458872i \(-0.151746\pi\)
0.888502 + 0.458872i \(0.151746\pi\)
\(854\) 0 0
\(855\) 5.86030 0.200418
\(856\) 0 0
\(857\) −32.2843 −1.10281 −0.551405 0.834238i \(-0.685908\pi\)
−0.551405 + 0.834238i \(0.685908\pi\)
\(858\) 0 0
\(859\) −28.7444 −0.980746 −0.490373 0.871513i \(-0.663139\pi\)
−0.490373 + 0.871513i \(0.663139\pi\)
\(860\) 0 0
\(861\) 16.0000 0.545279
\(862\) 0 0
\(863\) 43.0781 1.46640 0.733198 0.680015i \(-0.238027\pi\)
0.733198 + 0.680015i \(0.238027\pi\)
\(864\) 0 0
\(865\) 79.2548 2.69475
\(866\) 0 0
\(867\) −23.5181 −0.798718
\(868\) 0 0
\(869\) −29.2548 −0.992402
\(870\) 0 0
\(871\) 5.86030 0.198569
\(872\) 0 0
\(873\) 41.4558 1.40307
\(874\) 0 0
\(875\) 8.65914 0.292732
\(876\) 0 0
\(877\) −18.4437 −0.622798 −0.311399 0.950279i \(-0.600798\pi\)
−0.311399 + 0.950279i \(0.600798\pi\)
\(878\) 0 0
\(879\) −8.92177 −0.300924
\(880\) 0 0
\(881\) 27.9411 0.941360 0.470680 0.882304i \(-0.344009\pi\)
0.470680 + 0.882304i \(0.344009\pi\)
\(882\) 0 0
\(883\) −2.98454 −0.100438 −0.0502190 0.998738i \(-0.515992\pi\)
−0.0502190 + 0.998738i \(0.515992\pi\)
\(884\) 0 0
\(885\) −23.3137 −0.783682
\(886\) 0 0
\(887\) 10.1899 0.342142 0.171071 0.985259i \(-0.445277\pi\)
0.171071 + 0.985259i \(0.445277\pi\)
\(888\) 0 0
\(889\) 20.6863 0.693796
\(890\) 0 0
\(891\) 27.8477 0.932933
\(892\) 0 0
\(893\) −3.31371 −0.110889
\(894\) 0 0
\(895\) −5.86030 −0.195888
\(896\) 0 0
\(897\) 8.97056 0.299518
\(898\) 0 0
\(899\) −33.8937 −1.13042
\(900\) 0 0
\(901\) 20.9706 0.698631
\(902\) 0 0
\(903\) 10.4525 0.347838
\(904\) 0 0
\(905\) −19.6569 −0.653416
\(906\) 0 0
\(907\) −9.47890 −0.314742 −0.157371 0.987540i \(-0.550302\pi\)
−0.157371 + 0.987540i \(0.550302\pi\)
\(908\) 0 0
\(909\) −11.2132 −0.371918
\(910\) 0 0
\(911\) 16.0502 0.531766 0.265883 0.964005i \(-0.414337\pi\)
0.265883 + 0.964005i \(0.414337\pi\)
\(912\) 0 0
\(913\) −16.7696 −0.554991
\(914\) 0 0
\(915\) 116.617 3.85524
\(916\) 0 0
\(917\) −13.9411 −0.460377
\(918\) 0 0
\(919\) −35.4244 −1.16854 −0.584272 0.811558i \(-0.698620\pi\)
−0.584272 + 0.811558i \(0.698620\pi\)
\(920\) 0 0
\(921\) −67.1127 −2.21144
\(922\) 0 0
\(923\) −7.01962 −0.231054
\(924\) 0 0
\(925\) 33.7574 1.10994
\(926\) 0 0
\(927\) −10.7151 −0.351931
\(928\) 0 0
\(929\) −37.1716 −1.21956 −0.609780 0.792571i \(-0.708742\pi\)
−0.609780 + 0.792571i \(0.708742\pi\)
\(930\) 0 0
\(931\) 2.08786 0.0684269
\(932\) 0 0
\(933\) −18.6274 −0.609834
\(934\) 0 0
\(935\) −46.1396 −1.50893
\(936\) 0 0
\(937\) −7.45584 −0.243572 −0.121786 0.992556i \(-0.538862\pi\)
−0.121786 + 0.992556i \(0.538862\pi\)
\(938\) 0 0
\(939\) −22.6984 −0.740733
\(940\) 0 0
\(941\) −17.0711 −0.556501 −0.278250 0.960509i \(-0.589755\pi\)
−0.278250 + 0.960509i \(0.589755\pi\)
\(942\) 0 0
\(943\) 23.4412 0.763351
\(944\) 0 0
\(945\) 11.3137 0.368035
\(946\) 0 0
\(947\) 37.0321 1.20338 0.601691 0.798729i \(-0.294494\pi\)
0.601691 + 0.798729i \(0.294494\pi\)
\(948\) 0 0
\(949\) −6.14214 −0.199382
\(950\) 0 0
\(951\) −32.8882 −1.06647
\(952\) 0 0
\(953\) 44.9706 1.45674 0.728370 0.685184i \(-0.240278\pi\)
0.728370 + 0.685184i \(0.240278\pi\)
\(954\) 0 0
\(955\) −46.1396 −1.49304
\(956\) 0 0
\(957\) 57.2548 1.85079
\(958\) 0 0
\(959\) 32.1003 1.03657
\(960\) 0 0
\(961\) 23.6274 0.762175
\(962\) 0 0
\(963\) 38.3002 1.23421
\(964\) 0 0
\(965\) 56.2843 1.81185
\(966\) 0 0
\(967\) 4.59220 0.147675 0.0738376 0.997270i \(-0.476475\pi\)
0.0738376 + 0.997270i \(0.476475\pi\)
\(968\) 0 0
\(969\) 3.31371 0.106452
\(970\) 0 0
\(971\) 8.58221 0.275416 0.137708 0.990473i \(-0.456026\pi\)
0.137708 + 0.990473i \(0.456026\pi\)
\(972\) 0 0
\(973\) 29.3726 0.941642
\(974\) 0 0
\(975\) −10.1899 −0.326337
\(976\) 0 0
\(977\) 15.1127 0.483498 0.241749 0.970339i \(-0.422279\pi\)
0.241749 + 0.970339i \(0.422279\pi\)
\(978\) 0 0
\(979\) 3.95815 0.126503
\(980\) 0 0
\(981\) −4.10051 −0.130919
\(982\) 0 0
\(983\) −2.05600 −0.0655762 −0.0327881 0.999462i \(-0.510439\pi\)
−0.0327881 + 0.999462i \(0.510439\pi\)
\(984\) 0 0
\(985\) 50.2843 1.60219
\(986\) 0 0
\(987\) −29.5641 −0.941037
\(988\) 0 0
\(989\) 15.3137 0.486948
\(990\) 0 0
\(991\) −23.4412 −0.744635 −0.372317 0.928106i \(-0.621437\pi\)
−0.372317 + 0.928106i \(0.621437\pi\)
\(992\) 0 0
\(993\) 14.8284 0.470566
\(994\) 0 0
\(995\) 76.6004 2.42840
\(996\) 0 0
\(997\) 8.38478 0.265549 0.132774 0.991146i \(-0.457611\pi\)
0.132774 + 0.991146i \(0.457611\pi\)
\(998\) 0 0
\(999\) 10.9778 0.347321
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1024.2.a.h.1.4 4
3.2 odd 2 9216.2.a.bp.1.3 4
4.3 odd 2 inner 1024.2.a.h.1.1 4
8.3 odd 2 1024.2.a.i.1.4 4
8.5 even 2 1024.2.a.i.1.1 4
12.11 even 2 9216.2.a.bp.1.4 4
16.3 odd 4 1024.2.b.g.513.2 8
16.5 even 4 1024.2.b.g.513.1 8
16.11 odd 4 1024.2.b.g.513.7 8
16.13 even 4 1024.2.b.g.513.8 8
24.5 odd 2 9216.2.a.w.1.1 4
24.11 even 2 9216.2.a.w.1.2 4
32.3 odd 8 512.2.e.j.129.1 yes 8
32.5 even 8 512.2.e.i.385.1 yes 8
32.11 odd 8 512.2.e.j.385.1 yes 8
32.13 even 8 512.2.e.i.129.1 8
32.19 odd 8 512.2.e.i.129.4 yes 8
32.21 even 8 512.2.e.j.385.4 yes 8
32.27 odd 8 512.2.e.i.385.4 yes 8
32.29 even 8 512.2.e.j.129.4 yes 8
96.5 odd 8 4608.2.k.bi.3457.3 8
96.11 even 8 4608.2.k.bd.3457.2 8
96.29 odd 8 4608.2.k.bd.1153.2 8
96.35 even 8 4608.2.k.bd.1153.1 8
96.53 odd 8 4608.2.k.bd.3457.1 8
96.59 even 8 4608.2.k.bi.3457.4 8
96.77 odd 8 4608.2.k.bi.1153.4 8
96.83 even 8 4608.2.k.bi.1153.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.e.i.129.1 8 32.13 even 8
512.2.e.i.129.4 yes 8 32.19 odd 8
512.2.e.i.385.1 yes 8 32.5 even 8
512.2.e.i.385.4 yes 8 32.27 odd 8
512.2.e.j.129.1 yes 8 32.3 odd 8
512.2.e.j.129.4 yes 8 32.29 even 8
512.2.e.j.385.1 yes 8 32.11 odd 8
512.2.e.j.385.4 yes 8 32.21 even 8
1024.2.a.h.1.1 4 4.3 odd 2 inner
1024.2.a.h.1.4 4 1.1 even 1 trivial
1024.2.a.i.1.1 4 8.5 even 2
1024.2.a.i.1.4 4 8.3 odd 2
1024.2.b.g.513.1 8 16.5 even 4
1024.2.b.g.513.2 8 16.3 odd 4
1024.2.b.g.513.7 8 16.11 odd 4
1024.2.b.g.513.8 8 16.13 even 4
4608.2.k.bd.1153.1 8 96.35 even 8
4608.2.k.bd.1153.2 8 96.29 odd 8
4608.2.k.bd.3457.1 8 96.53 odd 8
4608.2.k.bd.3457.2 8 96.11 even 8
4608.2.k.bi.1153.3 8 96.83 even 8
4608.2.k.bi.1153.4 8 96.77 odd 8
4608.2.k.bi.3457.3 8 96.5 odd 8
4608.2.k.bi.3457.4 8 96.59 even 8
9216.2.a.w.1.1 4 24.5 odd 2
9216.2.a.w.1.2 4 24.11 even 2
9216.2.a.bp.1.3 4 3.2 odd 2
9216.2.a.bp.1.4 4 12.11 even 2